Graphics Reference
In-Depth Information
(
)
=+- +=
2
2
Cx y
,
4 8 5 40
x
x
y
.
25 16
The next three design problems continue to decrease the number of specified
points and increase the number of specified lines. Rather than solving them directly
as we did for the first two problems, we shall note the duality of points and lines in
projective space (as shown by equation (3.16) in Section 3.4) and essentially get our
new solutions from those above using this duality. There will be little new that has to
be proved. Mainly, we have to translate facts about points appropriately.
Definition.
A
line conic
in the projective plane is a set of lines
L
that satisfy the
equation
T
a
Q
= 0,
where [
L
] = [
a
] and Q is a symmetric 3 ¥ 3 matrix.
3.6.1.10. Theorem.
The set of tangent lines
L
to the nondegenerate point conic
defined by
T
p
Q
= 0
is the line conic defined by
-
1
T
aa
Q
=
0,
where [
L
] = [
a
].
Proof.
This theorem is an easy consequence of Theorem 3.6.1.5. We have replaced
the point [
p
] by the line [
a
], where
a
=
p
Q, which is tangent to the conic at
p
. Turning
this around, the point [
a
Q
-1
] corresponds to the line [
a
].
Conic design problem 3:
To find the equation of the conic that passes through two
points and is tangent to three lines, such that one of the lines passes through the first
point, another passes through the second point, and the third line is arbitrary. The
lines are not allowed to be concurrent and the neither of the first two lines can contain
both points. See Figure 3.27(a).
Solution.
We dualize the solution to design problem 2. Assume that the conic passes
through
p
1
and
p
2
and has tangent lines
L
1
and
L
2
at those points. Let
L
3
be the other
tangent line. Let
p
3
be the intersection of
L
1
and
L
2
, and let
p
i
have homogeneous
coordinates [x
i
,y
i
,z
i
]. Define symmetric 3 ¥ 3 matrices Q
1
and Q
2
by
1
2
(
)
T
T
T
(
) (
)
+
(
) (
)
=
(
) (
)
Q
=
xyz
,
,
xyz
,
,
xyz
,
,
xyz
,
,
and Q
xyz
,
,
xyz
,
,
.
1
111
222
222
111
2
333
333
Let [
L
3
] = [a
3
,b
3
,c
3
]. If
(
)
QQ
=+-
l
1
l
Q
,
l
1
2
